The Giants are said to be the other club, offering two pitching prospects. But San Francisco is looking for Miami to assume at least a portion of what is left on the right-hander's contract.
The Marlins have had conversations with more than a half dozen teams regarding Nolasco. They may be waiting to see if any more tempting offers surface.
In the final year of a three-year, $26.5 million contract, Nolasco is eligible for free agency after the season.
The 30-year-old is 5-8 with a 3.85 ERA in 18 starts. He's logged 112 1/3 innings with 90 strikeouts and 25 walks.
Nolasco is Miami's all-time leader in wins (81), innings pitched (1,225 2/3), games started (197) and strikeouts (1,001). He's coming off an impressive victory in Atlanta on Wednesday, giving up two runs on six hits in seven innings. He struck out seven and became the first pitcher in franchise history to reach 1,000 strikeouts.
On Friday, Nolasco threw his scheduled between starts bullpen session, and he is scheduled to pitch for the Marlins on Monday against the Braves at Marlins Park.
Miami is expected to deal Nolasco by the July 31 non-waiver Trade Deadline.
While Nolasco is a likely candidate to be dealt, the Marlins have no intention of dealing slugger Giancarlo Stanton, who is eligible for arbitration in 2014. The 23-year-old won't have the necessary service time to become a free agent until after the 2016 season.
Miami has been receiving calls on relievers Steve Cishek and Mike Dunn. But neither is expected to be dealt. The two are arbitration eligible next year, and they are seen as core pieces for the future.
But reliever Ryan Webb, in his first season of arbitration, is a candidate to be traded. Miami has been in conversations with at least a couple of teams regarding Webb.